Identities of nonterminating series by Zeilberger's algorithm

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

ELSEVIER

Journal of Computational and Applied Mathematics 99 (1998)449-461

Identities of nonterminating series by Zeilberger's algorithm Tom H. Koomwinder* Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, Netherlands Received 17 February 1998

Abstract

This paper argues that automated proofs of identities for nonterminating hypergeometric series are feasible by a combination of Zeilberger's algorithm and asymptotic estimates. For two analogues of Saalschiitz' summation formula in the nonterminating case this is illustrated. (~) 1998 Elsevier Science B.V. All rights reserved.

Keywords: Zeiberger's algorithm; Nonterminating hypergeometric series; Saalschiitz' summation formula; Computer algebra; Asymptotics

1. Introduction

The Gosper algorithm and in particular the subsequent Zeilberger algorithm and related WZ method have been extremely successful for an approach by computer algebra to identities involving (q-) hypergeometric functions (see the book by Petkov~ek et al. [7] and the references given there). Most of the current theory and all implementations of Zeilberger's algorithm remain restricted to the case of identities for terminating hypergeometric series, while many of the known identities in the literature involve nonterminating hypergeometric identities. The book [7] discusses in Ch. 7 some nonterminating cases obtained from terminating cases by the WZ method. A very convincing demonstration that Zeilberger's algorithmic is suitable for nonterminating cases is given by Gessel [3, p. 547]. He demonstrates Gauss' summation formula for the Gaussian hypergeometric series of argument 1 by means of a combination of Zeilberger's algorithm and the asymptotic estimate

F(a+k)~k~_t,

r(b + k)

ask---~

(see, for instance, [6, Ch. 4, Section 5.1] for a proof of (1.1)). * E-mail: [email protected]. 0377-0427/98/S-see front matter (~) 1998 Elsevier Science B.V. All rights reserved. PII: S0377-0427(98)00176-9

(1.1)

T.H. Koornwinder/ Journalof Computationaland AppliedMathematics99 (1998) 449-461

450

I will demonstrate in this paper that the method extends to the nonterminating generalization of Saalschfitz' summation formula for a terminating Saalschiitzian 3F2 hypergeometric series of argument 1. This nonterminating case has the form A + B = C, where A is a nonterminating Saalschfitzian 3F2(1), B is a quotient of Gamma functions times another 3F2(1) and C is a quotient of Gamma functions. Without knowing the formula explicitly, one can start with A, apply Zeilberger's algorithm to the terms of A, and finally arrive by some limit transition using (1.1) at the desired formula

A+B=C. It is important to notice that all steps in the proof, both in the Gauss case and in the nonterminating Saalschfitz case, may be automated, including the application of the asymptotic formula (1.1). I strongly believe that this idea may lead to an algorithmic approach to most identities for nonterminating hypergeometric series in literature. For the q-analogues of such identities an algorithmic approach should be feasible as well. For an introduction to Zeilberger's and related algorithms I refer to the book [7]. For hypergeometric series see Erdrlyi [1] and Gasper and Rahman [2]. Although the emphasis in [2] is on the q-case, it also contains some information on the q = 1 case. Anyhow, many formulas for q = 1 can be looked up from the corresponding q-case in [2], by silently taking the formal limit for ql"l, and by using that

Fq(z) :=

(1-q)l-'-(q;q)~ --~F(z) (qz;q)~

as qT 1

(see [5, Appendix B] and references given there). The contents of this paper are as follows. Section 2 discusses Gauss' summation formula following Gessel [3, p. 547]. Section 3 discusses the case of a nonterminating Saalschfitzian 3F2(1). Section 4 gives three other identities related to the one derived in Section 3, and it is shown how these identities follow analytically from each other. Finally, in Section 5, we consider one of the other identities in Section 4 (A + B = C with C a Gamma quotient and A and B a Gamma quotient times a nonterminating 3F2(1) with one of the upper parameters equal to 1). As is demonstrated, it can be derived by computer algebra and asymptotics, similarly as for the case of Section 3, but the asymptotics is quite tricky (maybe interesting in its own right) and not yet suitable to be automated.

2. Gauss' summation formula Gauss' summation formula (Gauss, 1813; see for instance [1, 2.8(46) with proof in Section 2.1.3]) is the nonterminating analogue of the Chu-Vandermonde summation formula. It reads as follows:

2Fl [a'cb; 11 := ~-~ (a)k(b)k k=0

(c)kk!

F(c-- a)F(c- b)'

(2.1)

where we assume that Re(c - a - b) > 0 and c ~ {0, - 1, - 2 , . . . } in order to ensure that the hypergeometric series on the left-hand side is well defined and absolutely convergent. Gessel [3, Section 7] shows how to prove this identity by the WZ method, as I will recapitulate now.

T.H. Koornwinder l Journal of Computational and Applied Mathematics 99 (1998) 449~t61

451

Formula (2.1), with c replaced by c + n (nE {0, 1,2 .... }), can be written as ~ ~ . f ( n , k ) =s(n),

(12.2)

k=O

where f(n,k).-

(a)k(b)k (c+n)kk!'

F(c + n)F(c + n - a - b) s(n) := F ( c + n - a ) F ( c + n - b)"

(2.3)

The sum on the left-hand side of (2.2) still absolutely converges (since n>~0). Formula (2.2) can be rewritten as ~-~F(n,k) = 1, k

(2.4)

0

where F(n,k):=

f(n,k) F(a + k)F(b + k ) r ( c + n - a)F(c + n - b) s(n) = F ( l + k ) F ( c + n + k ) F ( a ) F ( b ) F ( c + n - a - b ) "

(2.5)

By (1.1) we have F ( n , k ) ~ c o n s t . k a+h-C 1-n

as k---~oc.

Thus the sum on the left-hand side of (2.4) converges absolutely (as we already knew). We want to prove (2.4) by the WZ method. Gosper's algorithm applied to F(n + 1, k ) - F ( n , k ) or, equivalently, the WZ method applied to F ( n , k ) succeeds. In Maple V4, for instance, call read 'hsum.mpl': gosper(F(n+l,k)-F(n,k),k); or read 'hsum.mpl': WZcertificate(F(n,k),k,n);

from Koepf's package hsum.mpl [4], or call read ekhad; c t ( F ( n , k ) , l , k , n , N ) ; from Zeilberger's package EKHAD [8]. The resulting formula is F(n + 1,k) - F ( n , k ) = G ( n , k

÷ 1 ) - G(n,k),

(2.6)

where G(n,k) :=

k c+n

- a -

bF(n,k).

Note that

G(n, O) = O,

G(n, k) ~ const, k "+b ~ "

as k ---+oo.

(2.7)

T.H. Koornwinder/Journalof Computationaland Applied Mathematics 99 (1998) 449-461

452

It follows from (2.6) and (2.7) that K

K

F(n + 1, k) - ~ F(n, k) = G(n, K ÷ 1) - G(n, O) = G(n, K + 1) ---,0 as K ---, ec. k=0

k=0

Hence ~k~oF(n,k) is independent of n. Thus, OC

OO

OO

~C

~-'~F(n,k)= ,,~lim ~ _ F ( n , k ) = ~--~ ( limn+~F(n,k)) = ~ 6k.0 = 1, k=0

k=0

k=0

(2.8)

k=0

where the second equality is justified by dominated convergence. Indeed, for some C > 0 we have 1~Is(n)[ <~ C if n C {0, 1,2 .... } (use (2.3) and (1.1)). Thus, by (2.5) and (2.3) we obtain !a)~(b)k

[F(n,k)[ ~ C[f(n,k)[ <. C (Rec ÷ no)kk!

(2.9)

for nE{no, no+ 1,...}, where no is such that Rec + n0>0. Again by (1.1), the right-hand side of (2.9) summed over k = 0, 1,2,... yields a convergent sum, thus justifying the application of the dominated convergence theorem in (2.8).

3. Nonterminating analogue of Saalschiitz' summation formula Saalsch/itz' summation formula (which actually goes back to Pfaff, 1797) for a terminating Saalschfitzian 3F2 series with argument 1 reads as follows: 3F2

I

- m , b, c

l.e,-m+b+c-e+

1 ; 1 ::

1

k=0

(e - b)m(e - c)m (e)m(e - b - C)m (mE{O, 1 , 2 , . . . } , e , e - b - c ~ { O , - 1 , - 2 , . . . } ) .

(3.1)

See, for instance, [1, 4.4(3) with proof in Section 2.1.5]. Now replace - m by a on the left-hand side of (3.1), where a is arbitrarily complex: 3F2

[ a,b,c [e,a+b+c-e+

; 1 1

1 ~:= -~

(a)k(b)k(C)k k=0 ( e ) k ( a + b + c - e + 1)kk!'

(3.2)

and try to generalize identity (3.1) for this case. Such generalizations, with one additional 3F2(1) term, are known in literature, but we want to find a formula of this type from scratch, just starting from (3.2) and working with the Zeilberger algorithm. In (3.2) replace c by c + n, where n E {0, 1,2 .... }. Then 3F2

F

ke, a + b + c - e + n +

; 1 1

f(n,k),

= k=0

(3.3)

T.H. Koornwinder / Journal of Computational and Applied Mathematics 99 (1998) 449~461

453

where f(n,k):=

(a)k(b)k(c ÷ n)k (e)k(a + b + c - e + 1 + n)kk!'

f(n,k)~

F(e)F(a+b+c-e+

SO

1 ÷n'k-2)

as k---~ oo

r(a)F(b)r(c + n) by (1.1). Thus we have absolute convergence in the sum in (3.3). Zeilberger's algorithm applied to f ( n , k ) succeeds. In Maple V4, for instance, call read ekhad: ct(£(n,k),l,k,n,N);

from Zeilberger's package EKHAD [8]. The resulting recurrence is (c-e+a+n+

1)(b+c-e+n+

1 ) f ( n + 1,k)

- ( c - e + n + 1)(b + c - e + a + n ÷ 1 ) f ( n , k ) = g ( n , k + 1) - g(n,k),

(3.4)

where g(n, k) :=

1)(e + k -

(b+c-e+a+n+

1)k f ( n , k )

c+n

We can rewrite (3.4) in the form F(n + 1 , k ) - F ( n , k ) = G(n,k + 1 ) - G ( n , k ) ,

(3.5)

where F(a + c - e + 1 + n)F(b + c - e + 1 + n ) f ( n , k ) , F(c----eTl +n)F(a+b+c e + l + ;5 k(e + k - 1) G(n,k) := (c ÷ n)(c - e + n + 1) F(n'k)"

F(n,k):=

Observe that F(e)F(a+c-e+ l +n)F(b+c-e+ l +n)k F(a)F(b)F(c + n)F(c - e + 1 + n)

F(n,k)

2

as k --, oc,

and G(n,O)=O,

lim G(n,k)= k---+ o c

F(e)F(a + c - e + 1 ÷ n ) F ( b + c - e + 1 + n) F ( a ) F ( b ) r ( c + 1 + n ) r ( c - e + 2 + n)

(3.6)

Put ~c

S(n) := Z F(n, k) k=0

F(a+c-e+ 1 +n)F(b+c-e+ F(c-e+ 1 +n)F(a+b+c-e+

1 +n) 1 +n)

a,b,c + n 3F2

] ; 1

le, a + b + c - e + l + n

.

(3.7)

T.H. Koornwinder l Journal of Computational and Applied Mathematics 99 (1998) 449-461

454

Then by (3.5) and (3.6) we obtain

S(n + 1 ) - S(n)= k lim G(n,k + I ) - G(n,O) ~ cx~ F(e)F(a + c - e + 1 ÷ n)F(b + c - e + 1 ÷ n) F(a)F(b)F(c + 1 + n)F(c - e + 2 + n)

(3.8)

Hence, F(e)F(a + c - e + 1)F(b + c - e + 1) '~. (a+ c - e + 1)k(b + c - e + 1)k S(n)=S(O)+

F~F~-)ff(~;

-1) ~

Z e ~ .~~-

,=o

-(c ~- ? ~ - - -

e ;2-~ (3.9)

Note that

(a+c-e+

1)k(b+c-e+

1)k

F(c+ 1)F(c-e+2) .+b-e-I F(a + c - e + 1)F(b + c - e + 1) k

(c + 1 )k(c - e + 2)k

as k ---+~ .

Therefore assume that R e ( e - a - b ) > 0 . Then, for n ~ oc, the right-hand side of (3.9) tends to

F(a+c-e+ 1 ) F ( b + c - e + 1) F ( c - e + 1)F(a + b + c - e + 1)

a, b, c e,a+b+c-e+l

3F2

F(e)F(a + c - e + 1)F(b + c - e + 1) F(a)F(b)F(c + 1 ) r ( c - e + 2)

; 11

a+c-e+l,b+c-e+l,1

1 ; 1

3F2

.

c+ 1,c-e+2

Here we substituted the right-hand side of (3.7) with n = 0 for S(0). Next let n ~ e~ on the left-hand side of (3.9). From (3.7) we obtain

lim S(n)

= 2FI

a,b,

•

1

]

F ( e ) F ( e - a - b) b)

= F(e - a)F(e

(Re(e- a - b)>O).

(3.10)

e

The second identity is Gauss' summation formula (2.1). The first identity follows by applying (1.1) to the limit of the quotient of Gamma functions in (3.7) combined with a formal limit (to be justified in the lemma below) within the 3F2(1) in (3.7). Thus the limit case of (3.9) for n--~ ec is the identity we looked for:

F(a+c-e+ 1)F(b+c-e+ F(c-e+ 1)F(a+b+c-e+

1) 1)

a, b, c

] ; 1

3F2

e,a+b+c-e+l

F(e)F(a + c - e + 1)F(b + c - e + 1) F(a)F(b)F(c + 1)F(c - e + 2)

32[a+ c - e c++l ,lb, c+-ce-+e2+ l , 1 ;

1]

T. H. Koornw&der / Journal of Computational and Applied Mathematics 99 (1998) 449-461

F(e)F(e - a - b) F(e - a)F(e - b)'

455

(3.11)

valid for R e ( e - a - b ) > 0 . Let us now give the promised lemma which will justify the first identity in (3.10).

Lemma

3.1. Let R e ( e - a - b ) > 0 .

I

a,b,c+n

a, b, ]

]

lim 3 ~

"~

Then

; 1

Le, a + b + c _ e + l +

= 2FI

1 .

n

e

P r o o f . The limit formally holds because

(a)k(b)k(e + n)~ _ (a)k(b)k , , ~ (e)k(a + b + c - e + 1 + n)kk! (e)kk! lim

We will justify this limit b y dominated convergence. Take e E (0, 1 ) such that also e < Re(e - a - b). It follows from (1.1) that

(a),(b)k(e + n)k (e)k(a + b + c - e + 1 + n),k! F(a + k)F(b + k) = c o n s t . F(e + k ) F ( 1 + k )

×

F(c+a+b-e+l+n) F(c+n)

F(c+n+k) k) F(c+a+b-e+ 1 +n+

~< const, k Re("+h-') lnRe(a+h-e)+~(n + k) Re/e a-h)-t

= const, ( nn)-'~- ~ k

,_~(n_ 1 + k_,)ke I. . . . b)-~:

const.k -Rele-~-b)-~ ~< const.k .... J ~< const, k . . . .

t

if k<~n,

if k >~n. ~3G

Then the dominated convergence follows because ~t.=~ k . . . . ~
4. Other nonterminating identities related to Saalschiitz' summation formula Formula (3.11), which I derived in the previous section by a mixture o f computer algebra and asymptotic techniques, is possibly not in the literature, but it can be derived from some formulas o f similar nature which are in the literature.

456

T.H. Koornwinder / Journal of Computational and Applied Mathematics 99 (1998) 449~461

The best known nonterminating generalization of Saalsch/itz' formula (3.1) is

a, b,c

3F2

; 1

Le, a + b + c - e +

1

1

+ F(e-1)F(a-e+

l)F(b-e+

l)F(c-e+

l)F(a+b+c-e+

l)

F(1 - e)F(a)F(b)F(c)F(a + b + c - 2e + 2)

Ia-e+l,b-e+l,c-e+l

]

x3~

;1

2-e,a+b+c-2e+2 F ( a - e + 1)F(b- e + 1)F(c- e + 1)F(a + b + c - e + 1) F(1 - e ) F ( b + c - e + 1 ) F ( a + c - e + 1 ) F ( a + b - e + 1)'

(4.1)

see [2, (II.24)]. Since both 3F2(1 ) series are Saalschfitzian, there are no conditions on the parameters necessary for convergence. The second formula we will use involves two nonterminating 3F2(1 ) series with one upper parameter equal to 1, like the second 3F2(1) in formula (3.11 ). The formula reads as follows.

[a'b'l 1 3F2 / ;1 +

k d,e

3F2 [/ d - a ' b ' l ;1 1 1 kd, b _ e + 2

1-e

e-b-

F(d)F(e)F(a - e + 1)F(b - e + 1)F(d + e - a - b - 1) r(a)r(b)r(d

(Re(d + e - a -

b-

- a)F(d

(4.2)

- b)

1)>0,Re(a - e + 1)>0).

It can be derived by specialization of [2, (3.3.1)] combined with Gauss' summation formula (2.1). The third formula transforms a nonterminating Saalschfitzian 3F2(1 ) into a nonterminating 3F2( 1 ) with upper parameter 1:

a, b, c 3F2

Le, a + b + c - e +

] ; 11 =

1

F(e)F(a + b + c - e + 1) F(a)F(b + 1)F(c + 1) Ie-a,b+c-e+l,1 ×3~

] ; 1

(Rea>0).

(4.3)

b+l,c+l It follows by specialization o f [2, (3.1.2)]. Now replace in (4.3) a, b, c, e by b - e + 1, a - e + 1, c - e + 1,2 - e. Next substitute in the right-hand side of (4.3) (with parameters replaced as above) formula (4.2) with a, b,d,e replaced

T.H. Koornwinder / Journal of Computational and Applied Mathematics 99 (1998) 449~161 by 1 - b , a + c - e + l , a - e + 2 ,

c-e+2.

Ib-e+l,a-e+l,c-e+l

457

This yields

]

3F2

; 1

2 -e,a+b+c-2e+2 F(2 - e)F(a + b + c - 2e + 2) c F ( b - e + 1)F(a - e + 1 ) F ( c - e + 2)

l+b+c-e,l+a+c-e,

1 1 ; 1

×3F2 2 +c-e,c +

+ l

F(e - a - b)F(c)F(2 - e)F(a + b + c - 2e + 2) F(1 - b)F(1 + b + c - e)F(1 - a)F(a + c - e + 1)

(4.4)

(Re(1 + b - e ) > 0 , R e ( e - a - b ) > 0 ) . Now formula (3.11) follows from (4.1) by substituting (4.4) for the second using that r(z)r(1

-

z)

-

-

3F2(1) in (4.1), and by

-

sin(rtz)"

So we have seen that formulas (4.1)-(4.3) imply formula (3.11 ) via formula (4.4). Conversely, formulas (3.11), (4.2) and (4.3) imply formula (4.1) via formula (4.4). In the next section we will derive formulas (4.2) and (4.3) by methods of computer algebra and asymptotics.

5. Further proofs of identities of nonterminating series by computer algebra and asymptotics Let us try to prove formula (4.2) in a similar way as formula (3.11). Put

f(n,k) .-

(a)k(b + n)k (d + n)k(e)k"

By Zeilberger's algorithm we find that

(a - n - d)(n + b ) f ( n + 1, k) - (n + b - e + 1 )(n + d ) f ( n , k) = 9(n, k + 1 ) - .q(n, k), where

y(n,k) := - ( n + d)(e + k - 1 ) f ( n , k ) . Hence,

F(n + 1 , k ) - F ( n , k ) =

G(n,k + 1) - G(n,k),

T.H. Koornwinderl Journal of Computational and Applied Mathematics 99 (1998) 449-461

458

where

(b).(d - a). (a)k(b + n)k (d).(b - e + 1). (d + n)k(e)k' (e + k - 1)(a)k(b + n)k(b).(d - a). G(n,k):= ( e - n - b - 1)(d + n ) ~ ( e ) k ( d ) . ( b - e + 1)."

F(n,k) :=

Assume

Re(d + e - a - b -

1)>0,

(5.1)

Re(a - e + 1 ) > 0 .

Then

*--'F-n,k-=

(b).(d-a). a,b+n,l (d).(b - e + 1). 3F2 ; 1 d+n,e

S ( n ) : - - k=o

(5.2)

has absolutely convergent series and

S(n + 1) - S(n)= lim G(n,k + 1) - G(n,0) = - G ( n , 0 ) k---r o o

(e - 1)(d - a).(b). ( e - b - 1)(d).(b - e + 2)."

(5.3)

Hence,

s(o) = s(oo) +

e-1

~

(d-a),(b),

e - b - 1 ,=0 (d-~-~(b - - - e ~ 2 ) . '

where S(e~):=

lim S(n).

n~o(3

Thus, 3F2/

Ia l] el •

[ d,e

1

-

e

-

-

b

;1

3F21

1

[.d,b-e+2

]

+s(~).

So formula (4.2) will be proved if we can show that lim (b),(d-a), 3F2[a ' b + n ' l n~(d),(b-e+ l), [. d + n , e ;

F(d)F(e)F(a

-

e +

1 )F(b

r(a)r(b)r(d

[a1],

-

e +

1]

1 )F(d

- a)r(d

-

+ e -

a -

b -

1)

b)

This limit result is not evident. A formal limit would yield (lim

n e - a - l ) 2F,

;

e

=O-oc

since R e ( a - e +

1)>0.

(5.4)

459

T.H. Koornwinder l Journal of Computational and Applied Mathematics 99 (1998) 449-461

We will give a proof of (5.4) at the end of this section. Let us first turn to formula (4.3), and rewrite it in the following equivalent form:

[a,b+n, (b),(d-a), (d),(b-e+l),

1

3F2| [ d+n,e

r(d)F(e)F(b

]

; 1

- e + 1 )F(d + e - a - b - 1 )

F(b)r(d -

F ( b + n ) F ( d - a + n) F(b-e+

a ) F ( d + e - b - 1)

[d+n-l,d+e-a-b-l,e-1

× 3F2l

] ; 1J

l +n)F(d

+e-a-

l +n)

(Re(d + e - a - b - 1 ) > 0 ) .

(5.5)

[. d + e - a + n - l , d + e - b - 1

Observe that we can take limits on the right-hand side of (5.5) for n ~ e ~ (see (3.10)), and thus arrive at (5.4). However, I only want to use (5.5) if I can derive it by methods of computer algebra and asymptotics. Let us try. Write (5.5) more compactly as (5.6)

S(n) = CS(n),

where S ( n ) is given by (5.2), C:=

r(d)r(e)r(b

-

e +

1 ) r ( d + e - a - b - 1)

F(b)F(d - a)F(d + e - b - 1)

'

and S(n):=

F ( b + n ) F ( d - a + n)

[d+n-l,d+e-a-b-l,e-1

]

F ( b - e + 1 + n ) F ( d + e - a - 1 + n) 3F2 / L d+e-a+n-l,d+e-b-1

Note that S(n) is just (3.7) with a , b , c , e replaced by e Thus (3.8) yields that

1,d + e -

a-

b-

; 1J. 1,d-

l,d + e-

b-

1.

S(n + 1) - S(n) = 1 (e - 1)(d - a ) , ( b ) , C (b-e + l)(d),(b-e + 2),"

Comparison with (5.3) yields that S ( n ) - C S ( n ) is independent of n. Hence S ( n ) - C S ( n ) = S(cxz) - Cg(cx~).

Thus (5.6) holds iff (5.4) holds. So we have not made real progress. Formula (5.4) will follow from (5.6), but for the proof of (5.6) we need (5.4). Let us therefore give an independent proof of (5.4). P r o o f o f (5.4). Observe that

(b),(d-a), 3F2 ( d ) , ( b - e + 1 ),

a,b+n, 1; 1 1 d + n, e

T. H. Koornwinder / Journal of Computational and Applied Mathematics 99 (1998) 449-461

460

_F(d)F(e)F(b-e+l)~F(a+k)F(b+n+k) F(d-a+n) ~ ( ~ Z a ) k:o r ( e + k ) C ( d + n + k ) r ( b - e + 1 +n)"

(5.7)

We will first work formally, and consider the sum obtained by replacing in the above infinite sum the gamma quotients by their asymptotic estimates (for big n and k) as given by (1.1). This yields

-~ F ( a + k ) F ( b + n + k ) F(d-a+n) k:o r(e + k) r(d + n T k ) r ~ - e ~-f Un) ~

(k + 1 )~-"(n + k + 2)h-d(n + 1)d+e-~-h-I

(5.8)

(5.9)

k=O

_

~-~(

1

n+l

1 +

k=0

k + i)h-d ( k ÷ l ) " - " n+ \ n + lJ

(1 +t)~-ata-edt

(5.10)

as n---~oc

(5.11)

o

F ( a - e + 1)F(d + e - a - b - 1) r(d - b) Here we used that the integral in (5.11), which converges because of assumptions (5.1), is approximated by its Riemann sum (5.10) as n--~ cx~. Thus formally the right-hand side of (5.7) indeed equals the right-hand side of (5.4). We will justify this formal proof by Lebesgue's dominated convergence theorem. Rewrite (5.8) as

f ~ ?,(t)h,(t) dt,

(5.12)

where for k/(n + 1)< t <<.(k + 1)/(n + 1),

h,(t):=

Q

l+--

k÷llh-d(k÷l~"-" n+ \ n + lJ

and ?,(t) :=

F(a + k) F(b + n + k) F(d - a + n) F(e + k)(k + 1) ~-'~ F(d + n + k)(n + k + 2) b-J F(b - e + 1 ÷ n)(n + 1) d+e-a-h-~ "

Then Iv.(t)l ~< const,

and

[hn(t)l ~< const.(1 +

t)Re(b-d)t Re(a-e).

Furthermore, lim 7n(t)= 1 ~/~OC

and

lim h,(t)=(1 + t) b dta--e. n~OC

It follows by dominated convergence that the integral (5.12) tends to the integral (5.11) as n ~ cx~.

T.H. Koornwinder / Journal of Computational and Appfied Mathematics 99 (1998) 449-461

461

References [1] [2] [3] [4] [5] [6] [7] [8]

A. Erdrlyi et al., Higher Transcendental Functions, vol. I, McGraw-Hill, 1953 (reprinted in 1981 by R.E. Krieger). G. Gasper, M. Rahman, Basic Hypergeometric Series, Cambridge University Press, Cambridge, 1990. 1.M. Gessel, Finding identities with the WZ method, J. Symbolic Comput. 20 (1995) 537-566. W. Koepf, Maple package hsum.mpl, Version 1.0, February 1, 1998; obtainable by email from [email protected] eipzig.de. T.H. Koornwinder, Jacobi functions as limit cases of q-ultraspherical polynomials, J. Math. Anal. Appl. 148 (1990) 44 -54. F.W.J. Olver, Asymptotics and Special Functions, Academic Press, New York, 1974 (reprinted in 1997 by A.K. Peters). M. Petkov~ek, H.S. Wilf, D. Zeilberger, A =B, A.K. Peters, Wellesley, MA, 1996. D. Zeilberger, Maple package EKHAD, Version of 27 March 1997; obtainable from URL http://www.math.temple. edu/~ zeilberg.